Modern theories on quantum mechanics predict the significance of something called the "Planck Length". The Planck Length is equal to approximately 1.616253x10^-35 meters. That's almost unimaginably small; somewhere near ten sextillionths the length of a single proton (yes, a sextillionth is a real number, and as far as I know, not the name of a porno).
The Planck Length has significant mathematical significance in most commonly accepted quantum theories; it is one of several units of measurement (all named after Max Planck) that are based purely on perceived universal constants and not on measurements of physical objects. In particular, the Planck Length is derived from the strength of gravity, the speed of light, and the relationship between the energy of a photon and the frequency of its electromagnetic wave. Note that these three constants are all derived from separate physical theories: special relativity, general relativity, and quantum mechanics respectively. Relativity and quantum mechanics contradict each other in many ways (this is one of the greatest unsolved scientific puzzles of our time) so to me, the significance of the Planck Length is already suspect.
Many theories on the behavior of the universe at a very small scale (most notably several forms of a theory called quantum gravity) also state that the Planck Length has special physical significance. In fact, some physicists seem to have a preoccupation, or at least an interest, in determining the physical significance of the Planck Length. This is part of the point I will eventually be making. Some quantum theories state that relativity breaks down at distances less than the Planck Length and the universe at smaller scales behaves as sort of a multidimensional foam (don't ask me; I don't understand that one either). Some state that the universe may be quantized at this length, much as an image on a computer screen is quantized, or becomes blocky, at the length of one pixel. Some state that the Planck Length may have influence on the speed of light.
Scientists were recently able to test the latter.
The Fermi Gamma-Ray Space Telescope recently observed a gamma-ray burst called GRB 090423. Along with the normal smattering of high-energy photons, it picked up one that was truly exceptional. It came in at a whopping 31 giga-electron-volts. That's extraordinarily energetic. Energetic enough that its wavelength came to approximately 4x10^-17 meters. This was a far cry from the 1.6x10^-35 meters of the Planck Length, but it was just close enough so that the Planck Length's influence on its speed could potentially be measured when you consider the massive distance that the photons traveled. And what did they find?
When compared to the gamma-ray burst's other photons, it did not arrive at the predicted time differential.
Does this mean that quantum theory is wrong? Not at all. But it does mean that the physical significance of the Planck Length is highly suspect. To be fair, scientists are taking this a lot more seriously than the quasar problem, and are beginning to eliminate some forms of quantum gravity from theoretical models due to this discovery. However, they don't seem to be taking a second look at the big picture, which I believe suggests that mathematical models do an inadequate job of describing the nature of the universe.
Much of today's physics are based purely on mathematical models. All of string theory, for example, is based upon the idea that the behavior of subatomic particles seems to fit well with mathematical equations that describe the vibrations of multidimensional strings (hence the name). The science was born from the math. The Big Bang theory was based upon the perceived expansion of the universe, but many of the details that make up the theory are based purely on math. Originally, the Big Bang theory predicted a universe much older than it currently does. Then when observations showed otherwise, the underlying math of the theory was altered to fit the observations. This happened not once, but three or four times! The entire concept of the "inflationary phase" of the early universe was added to force the math behind the theory to fit the observations of the cosmic microwave background radiation.
This appears to be very different from the methods used to make discoveries in the past. They were made by coming up with an idea, then later finding math that fit that idea. Newton came up with the idea that all matter has a slight attraction to all other matter, then later discovered the math that became a successful model of gravity. Einstein came up with the idea that if you're traveling very fast and then shined a light ahead of yourself, the light is still traveling at the same speed as when you are standing still. Then later, he discovered the math that became a successful model of special relativity. Today's science is doing it backward; they are coming up with the math first, using the math as a theory of how the universe works, then changing the math when observed behavior doesn't fit the theory.
I suggest to you that this is a flawed method. I'm not saying that using this method can't come up with answers, but I believe that it will come up with a lot of mathematically-correct wrong answers (such as those rejected forms of quantum gravity) before it stumbles upon the right ones. Do the Planck Length and the other Planck units have mathematical significance? Probably. At the very least, they make it easier to express measurements on the levels needed for quantum theory. Do they have inherit physical significance? My guess is no.
Trip to Space